So I know that if you play a natural harmonic at the 12th fret, this is just the same note as fretting the 12th is. Also, when I play my electric guitar and listen to lots of metal music in particular, I can hear that some natural harmonics sound really good after certain chords etc. while others do not. Does anyone have any theory or a resource that could reveal the notes the natural harmonics along the fretboard produce because I really like using them in my playing but often just don't know if they're going to sound good or not at that time.

And if you can answer the same for pinch and touch harmonics, that would also be useful.

3 Answers
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Harmonics occur at the whole-number ratios of the length of the string, e.g. 1/2 the length of the string, 1/3 the length of the string, 2/3 the length of the string, 1/4, 3/4, etc. The lower the numbers involved, the easier the harmonic will be to play. So for example, theoretically there's a harmonic at 6/19 the length of the string, but it will be much harder to play than the one nearby at 1/3 the length of the string.

To determine the pitch of the harmonic, just divide the open string's frequency by the harmonic's string-length ratio. For example: the 7th fret is 1/3 of the length of the string (from the nut), so if the open string's frequency is X, then the frequency of the harmonic at the 7th fret is X/(1/3) = 3 X, which is an octave-and-perfect-fifth above X. Two harmonics who's string-length ratios have the same denominator (when in lowest terms) have the same pitch, so for example the harmonic at 2/3 the length of the string will also be an octave-and-perfect-fifth above X.

The most common harmonics occur at:

12th fret (ratio 1/2): an octave above the open string

7th fret (ratio 1/3): an octave plus a fifth above the open string

5th fret (ratio 1/4): two octaves above the open string

4th fret (ratio 1/5): two octaves plus a major third above the open string

9th fret (ratio 2/5; equivalent to 1/5): same as at the 4th fret, since the ratios have the same denominator.

Those are all I can think of right now; feel free to mention other common harmonics in the comments and I'll add them.

All of this works for pinch and touch harmonics as well, since when you fret a string, you're basically making it into an open string with a shorter length. The harmonics will all be in the same places relative to the length of the fretted string. In practical terms, to determine the fret at which a fretted string's harmonics occur, all you have to do is add the fret number to the harmonic's fret number. For example, if you fret a string at the 4th fret, then the "12th fret harmonic" will now occur at the 4 + 12 = 16th fret, so there will be a harmonic at the 16th fret with a pitch one octave higher than the fretted note.

5th fret harmonics are the same note as the open string, 7th fret harmonics are the same as the 7th fret notes (low to high - b,e,a,d,f#,b). Touch and tapped harmonics are the same as the fretted note assuming you are touching, tapping one octave higher and pinch harmonics change depending on where you pick/pinch the note. For pinch harmonics if you pick one or two octaves higher up the fretboard you will get the same note. You can usually get a clear harmonic a fifth (7frets) higher as well. eg press on the third fret and perform a pich harmonic on the 22nd fret. The harmonic will be a fifth higher than the fretted note. I would suggest getting a tuner and experimenting for yourself, its more fun that way and you might find something cool. Have fun.

Be aware, too, that the guitar is a tempered instrument. In order for all keys to sound close to in tune, some compromises are made, so that thirds for example are not exactly tuned to a whole number ratio. This means that the further up the harmonics go (the higher the numbers in the whole number ratios others describe), the less guarantee you have that the harmonic will match its tempered equivalent. Check out Musical Temperament on Wikipedia for a discussion.